Let `f(x)=xsinpix ,\ x >0` . Then for all natural numbers `n ,\ f^(prime)(x)` vanishes at (a) A unique point in the interval `(n ,\ n+1/2)` (b) a unique point in the interval `(n+1/2,\ n+1)` (c) a unique point in the interval `(n ,\ n+1)` (d) two points in the interval `(n ,\ n+1)`

Explanation

To solve the problem, we need to analyze the function \( f(x) = x \sin(\pi x) \) and its derivative \( f'(x) \) to determine where it vanishes for natural numbers \( n \).

### Step 1: Find the derivative of \( f(x) \)

Using the product rule, we differentiate \( f(x) \):

\[
f'(x) = \frac{d}{dx}(x) \cdot \sin(\pi x) + x \cdot \frac{d}{dx}(\sin(\pi x))
...